Categorification and mirror symmetry for Grassmannians

The homogeneous coordinate ring CGr (k, n) of the Grassmannian is a cluster algebra, with an additive categorification CMC. Thus every M has a cluster character M (k, n). The aim is to use the categorification to enrich Rietsch-Williams' mirror symmetry result that the Newton-Okounkov (NO) body/cone, made from leading exponents of functions in CGr (k, n) in an X-cluster chart, can also be described by tropicalisation of the Marsh-Reitsch superpotential~W. For any cluster tilting object T, with endomorphism algebra A, we define two new cluster characters, a generalised partition function PTM (CMA) and a generalised flow polynomial FTM (fdA), related by a `dehomogenising' map wt K (CMA) K (fdA). In the X-cluster chart corresponding to T, the function M becomes FTM and thus its leading exponent is (T, M), an invariant introduced in earlier paper (and the image of the g-vector of M under wt). When T mutates, FTM undergoes X-mutation and (T, M) undergoes tropical A-mutation. We then show that the monoid of g-vectors is saturated, and that this cone can be identified with the NO-cone, so the NO-body of Rietsch--Williams can be described in terms of (T, M). Furthermore, we adapt Rietsch-Williams' mirror symmetry strategy to find module-theoretic inequalities that determine the cone of g-vectors. Some of the machinery we develop works in a greater generality, which is relevant to the positroid subvarieties of Gr (k, n).